Hindawi Publishing CorporationFixed Point Theory and Applications Volume 2008, Article ID 418971, 4 pages doi:10.1155/2008/418971 Research Article T-Stability of Picard Iteration in Metr
Trang 1Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2008, Article ID 418971, 4 pages
doi:10.1155/2008/418971
Research Article
T-Stability of Picard Iteration in Metric Spaces
Yuan Qing 1 and B E Rhoades 2
1 Department of Mathematics, Beijing University of Aeronautics and Astronautics,
Beijing 100083, China
2 Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA
Correspondence should be addressed to Yuan Qing, yuanqingbuaa@hotmail.com
Received 10 July 2007; Accepted 11 January 2008
Recommended by H´el`ene Frankowska
We establish a general result for the stability of Picard’s iteration Several theorems in the literature are obtained as special cases.
Copyright q 2008 Y Qing and B E Rhoades This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
LetX, d be a complete metric space and T a self-map of X Let x n1 fT, x n be some
itera-tion procedure Suppose that FT, the fixed point set of T, is nonempty and that x nconverges
to a point q ∈ FT Let {y n } ⊂ X and define n dy n1 , fT, y n If lim n 0 implies that
lim y n q, then the iteration procedure x n1 fT, x n is said to be T-stable Without loss of
generality, we may assume that {y n } is bounded, for if {y n} is not bounded, then it cannot
possibly converge If these conditions hold for x n1 Tx n, that is, Picard’s iteration, then we
will say that Picard’s iteration is T-stable.
We will obtain sufficient conditions that Picard’s iteration is T-stable for an arbitrary
self-map, and then demonstrate that a number of contractive conditions are Picard T-stable.
We will need the following lemma from1
Lemma 1 Let {x n }, { n } be nonnegative sequences satisfying x n1 ≤ hx n n , for all n ∈ N, 0 ≤ h <
1, lim n 0 Then, lim x n 0.
Theorem 1 Let X, d be a nonempty complete metric space and T a self-map of X with FT / ∅ If
there exist numbers L ≥ 0, 0 ≤ h < 1, such that
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for each x ∈ X, q ∈ FT, and, in addition,
lim d
y n , Ty n
then Picard’s iteration is T-stable.
Proof First, we show that the fixed point q of T is unique Suppose p is another fixed point of
T, then
Since 0≤ h < 1, so dp, q 0, that is, p q.
Let{y n } ⊂ X, n dy n1 , Ty n , and lim n 0 We need to show that lim y n q.
Using1, 2, andLemma 1,
d
y n1 , q
≤ dy n1 , Ty n
dTy n , q
≤ n Ldy n , Ty n
hdy n , q
and lim y n q.
Corollary 1 Let X, d be a nonempty complete metric space and T a self-map of X satisfying the
following: there exists 0 ≤ h < 1, such that, for each x, y ∈ X,
dTx, Ty ≤ h max
dx, y, dx, Tx, dy, Ty, dx, Ty, dy, Tx
Then, Picard’s iteration is T-stable.
Proof From 2, Theorem 11, T has a unique fixed point q Also, T satisfies 1 It remains to show that2 is satisfied
Define p n to be the diameter of the orbit of y n ; that is, p n δOy n , Ty n , First, we show that p nis bounded:
d
Ty n , q
≤ h maxd
y n , q
, d
y n , Ty n
, d
y n , Tq
, d
q, Ty n
, d
q, Tq
≤ h maxd
y n , q
, d
y n , Ty n
, d
y n , q
, d
q, Ty n
, 0
h maxd
y n , q
, d
y n , Ty n
, d
y n , q
, d
q, Ty n
.
6
Hence, dTy n , q ≤ hdy n , q or dTy n , q ≤ hdy n , Ty n or dTy n , q ≤ hdq, Ty n
If dTy n , q ≤ hdy n , q, it is clear that
d
Ty n , q
≤ hdy n , q
1− h d
y n , q
If dTy n , q ≤ hdq, Ty n, then
d
Ty n , q
0 ≤ h
1− h d
y n , q
Trang 3Y Qing and B E Rhoades 3
If dTy n , q ≤ hdy n , Ty n, then
d
y n , Ty n
≤ dTy n , q
dy n , q
≤ hdy n , Ty n
dy n , q
Hence, dTy n , q ≤ h/1 − hdy n , q Now it is easy to see that {Ty n} is bounded and so is
{p n }, since {y n} is bounded
For any i, j ≥ n, using 5,
d
Ty i , Ty j
≤ h maxd
y i , y j
, d
y i , Ty i
, d
y j , Ty j
, d
y i , Ty j
, d
y j , Ty i
≤ hp n 10 Thus,
d
y i , Ty j
≤ dy i , Ty i−1
dTy i−1 , Ty j
≤ i−1 hp n−1 11 But
d
y i , y j
≤dy i , Ty i−1
dTy i−1 , Ty j−1
dTy j−1 , y j
≤ i−1 hp n−1 i−1 , 12 which implies that
and lim p n 0 byLemma 1 Since dyn , Ty n ≤ p n , lim dy n , Ty n 0
The conclusion now follows fromTheorem 1
Corollary 2 see 3, Theorem 1 Let X, d be a nonempty complete metric space and T a self-map
of X satisfying
for all x, y ∈ X, where L ≥ 0, 0 ≤ a < 1 Suppose that T has a fixed point p Then, T is Picard T-stable Proof Since T satisfies 14 for all x, y ∈ X, then T satisfies inequality 1 of our paper Let
{y n } ⊂ X and define n dy n1 , y n From the proof of Theorem 1 of 3, lim dyn , Ty n 0 Therefore, by our theoremTheorem 1, T is Picard T-stable
Definition5 of this paper is actually Definition 24 of 2 Therefore, many contractive conditions are special cases of 5, and, for each of these, Picard’s iteration is T-stable For example, Theorems 1 and 2 of4 and Theorem 1 of 5 are special cases ofCorollary 1
We will not examine the analogues ofTheorem 1for Mann, Ishikawa, Kirk, or any other iteration scheme since, if one obtains convergence to a fixed point for a map using Picard’s iteration, there is no point in considering any other more complicated iteration procedure
Acknowledgment
This article is partly supported by the National Natural Science Foundation of China no 10271012
Trang 44 Fixed Point Theory and Applications
References
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